David Clampitt

Analytical Applications of Singular Pairwise Well-formed Scale Structures

Pairwise well-formed (PWWF) scales are certain scales with an odd number of notes and three distinct step-interval sizes. Examples are found in world music generally as well as in Western music. Well-formed scales generalize diatonic scales; in the case of the usual diatonic, there are, of course, two step-interval sizes, with multiplicities 5 and 2. Almost all PWWF scales may be shown to have two distinct step-interval sizes with the same multiplicity, and a third step-interval size with a different (odd-number) multiplicity. The class of singular pairwise well-formed scales are set apart, with 7 notes, and step-interval sizes in three distinct multiplicities: 1, 2, and 4. Such scales have a symmetrical form <abacaba> (for example, C D-flat E F G A-flat B (C)), a structure that is reproduced for each of its generic interval cycles (cycles of 3rds/6ths, 4ths/5ths, as well as of 2nds/7ths). Such scales have a reciprocal relation to the usual diatonic: the defining equivalences, in turn, interval sizes a with b, b with c, and a with c, project to the three diatonic generic interval cycles.

The most abundant musical examples of the singular PWWF class are modes of the so-called Hungarian or gypsy minor (or otherwise identified scales in world music with this structure). One mode is the example above, a ‘major’ mode example, which admits a ‘minor’ mode dual. This scale is used in western art music, including explicitly in Brahms’s Violin Concerto and String Quintet, Op. 111, ii, and implicitly in Schubert’s Der Doppelgänger and Beethoven’s String Quartet, Op. 131, i. This paper will relate structural properties of singular PWWF scales to analytical insights into the Beethoven and Schubert examples (and suggest a historical relation between them), and to the slow movement of Brahms’s Op. 111.